When the children push the box with a force greater than the friction force, there is an unbalanced force, causing the box to start moving.
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The box still does not move because the friction force increases to balance the increased pushing force.
The friction force arises due to the interaction between the bottom surface of the box and the rough surface of the floor.
The box does not move because of friction acting in the opposite direction to the push, which balances the pushing force.
Unbalanced forces cause an object to move in the direction of the greater force, thus bringing the object into motion or changing its state of motion.